You can get some download manager applications from the app store, for what I know, those apps which can download videos from Internet are paid version, most of the free version are capable of downloading songs and files. Tap "download" or "download manager" in the search bar in the app store. Once you use a download manager to download a video, the downloaded video will be store in the download manager app, most of the time, there are few options allowing you to open the downloaded video using other capable applications.
If your phone is jailbreaked, then look for "safari download manager" in cydia. Safari download manager will store your downloaded videos in a folder called "download" in your system files, you will need another cydia app called "iFiles" or shh stuff to be able to view your downloaded videos.
